Luminance vs. Illuminance Relationship

Definition of Exposure

The altering of the silver halide grains by the photochemical action of the light in order to produce a latent image is called exposure.

To a first approximation (1) it has been found that the photochemical action taking place during exposure obeys the reciprocity law of Bunsen and Roscoe. As an application of this law to photochemical reactions, the exposure E may be expressed as

      E = It                                                  (1)  

where I = the intensity of the light acting upon the sensitized
          photographic material  
      t = the time during which this illumination is permitted to act on 
          the photographic material.  

For exposure to white light, which represents the usual condition, the exposure is measured in meter-candle-seconds. It is assumed that the light source has a spectral distribution like that of mean noon sunlight, and the exposure time t is usually continuous rather than the integrated effect of intermittent or chopped exposures.(2) The equation shows that the exposure, and consequently the photographic effect as measured by the density of silver deposit, depends directly upon the intensity of the light source and increases the longer the material is subjected to light rays. Because the exposure depends upon the time during which the light acts on the film, the film is able to integrate the quantity of light falling upon it. A practical advantage of this effect is that through sufficiently long exposure it is possible to photograph objects which might otherwise not be sufficiently bright to produce a photographic image.

The exposure is not the only factor determining the photographic effect produced, although it is a very important factor in this connection. The photographic effect, by which is meant the density of the silver deposit, is determined by the characteristics of the sensitive material and by the processing conditions as well as by the exposure.

These factors are related graphically by means of the D log10E characteristic and form a major component of the topic of Photographic Sensitometry.

By means of the D log10E characteristic curves, it is possible to determine the density produced on a certain photosensitive material for given exposure and processing conditions. Such curves provide a clue to what might be expected, by way of the photographic effect, when photographing an object with a camera and lens system.

However, the D-log10E characteristics are usually expressed in terms of density and meter-candle-seconds. The illumination of the object being photographed is not ordinarily determined in meter-candle-seconds, and even if this were possible, through the use of properly calibrated exposure or illumination meters, the intensity of illumination on the plate is vastly different from that of the or iginal object because of the reduction in size, the effect of the aperture stop, the focal length, and other characteristics of the lens system. To make maximum use of the sensitometric concepts and to understand fully the various and numerous factors which enter into exposure, it is desirable to provide the connecting link which relates the exposure, as given in the sensitometric sense of the term, and the brightness of the object as this may be determined by measurements with an exposure or illumination meter. It is proposed to construct this connecting link based upon theoretical considerations for two reasons: 1) An understanding of the theory of exposure provides an excellent basis for understanding the practical treatment which is to follow (2) the relations and equations which are derived from theoretical considerations are required for a full explanation of the use of exposure tables and light meters.

diagram of lens system]

FIG. 1. Optical system of a camera showing axial rays. The luminous intensity , I', of the point P' on the photographic plate can be expressed in terms of the luminous intensity, I, of the point on the subject, P, and the characteristics of the lens system.

Image Brightness as a Function of Optical System

We will now establish the connecting link by which the brightness of the image on the photographic plate may be determined from the illumination of the original object being photographed. This link involves the optical system of the camera, which, so far exposure is concerned, includes the iris diaphragm, the bellows extension, a filter (if one is used), and the shutter, as well as the lens system proper.

Let Fig. 1 represent the lens system of the camera, in which a point object P , whose luminous intensity is I, produces an image object of itself P' with luminous intensity I' on the photographic plate. The iris diaphragm or aperture is represented as being at A. The principal planes of the lens are represented as lying at PP and at PP', while the entrance and exit pupils are designated as being at NP and XP, respectively, and the principal focal lengths are L and L'.

The point P may be self-luminous or may be illuminated by reflected light. In either case it will illuminate the entrance pupil of the lens NP with an intensity inversely proportional to the square of the distance between P and R, the latter being in the plane of the entrance pupil, and directly proportional to its luminous intensity I.

Let the distance between P and R be X. Then the intensity of light falling up on the entrance pupil will be proportional to I/X². The distance X may be considered as being made up of two components. One of these is the distance from P to the interior principal focal length L, which distance is given by -L/M where M is the linear magnification produced by the lens system. The negative sign is required because of the inversion of the image. The second component of the distance X is the distance LR from the plane of the principal focus to the plane of the entrance pupil. Since the principal-focus and the entrance-pupil planes are never very far removed from one another, the distance LR may be expressed by L(1 + q) where q is a small positive or negative decimal. Neglecting the negative sign required because of the image inversion, the distance from P to R may be expressed as

         X = L[ (1 + q) + 1/M ]                               (2)  

The intensity I' of the light at the point P' is proportional to the cone whose half angle is q. The maximum diameter of this cone at the exit pupil is determined by the area of the aperture, which is given by

         A =  	π d²  /  4                                (3)  

where d = the diameter of the aperture.  
      	π = the value 3.1416

The cone of light emerging from the exit pupil comes to a focus at P' and produces an image of P whose size is proportional to the linear magnification of the system M. The intensity of the image at P' is inversely proportional to the area of the image. But the area of this image is

         a =  	π M²  /  4                                (4)  

so that I' is proportional to 4/ π M²

Finally, the intensity of the image at P' is reduced by absorption and reflection by the separate elements of the lens system. Of the light incident upon the lens, some is absorbed, but a greater part is reflected from the lens surfaces, especially if these are uncemented. The quantity of the emerging light is always less than that incident upon the system and is proportional to the incident light and the transmission of the lens system T. Consequently I' is proportional to T.

Having discussed briefly the separate factors which influence the intensity of the image, we may now combine the separate effects. Thus, for an object P on, or very near to, the optical axis of the lens system, the intensity of the image is

        I' = kIAT / X² a                                (5)  

where k is a numerical constant depending upon the units of measurement.

By substituting for A, X, and a, the values already determined, and by simplifying, the expression becomes

        I' = I [ kTd² / L² [M(1 + q) + 1]² ] ]     (6)  

Since q is a small fraction, little error is introduced if it is neglected, and for practical purposes the above equation may be simplified to

        I' = I [kT d² / L² (M+1)²]          (7)  

This equation gives in the most general form the connecting link relating the intensity of the image and that of the original subject, so far as the lens system is concerned, provided that the object and image are not far removed from the optical axis of the lens. For objects considerably off the optical axis, and especially when the view angle is large, the intensity of the image at a corner of the plate may vary considerably from that given by Eq. (7).

By definition the f-number of a lens is the ratio of the focal length to the diameter of the aperture. Thus we may substitute f for L/d in the above equations, where f represents the f-number of the lens for a specified diameter of aperture d. When this substitution is made, we obtain:

        I' = IkT / f² (M + 1)²                    (8)  

Which is, perhaps, in its simplest and most practical form the connecting link relating the intensity of the image and that of the original subject. This equation states that the intensity of the image is proportional to the intensity of the original object, proportional to the transmission of the lens system, inversely proportional to the square of the f number, and inversely proportional to the square of the linear magnification plus one.

Image Brightness as Function of Filter

A filter is frequently employed in photography to increase contrast, to produce desirable pictorial effects, or to distinguish between tone rendition of various colors. The property inherent in all filters is absorption of a portion of the spectrum to which the photographic emulsion is sensitive, and it thereby decreases the effective intensity of illumination on the photographic material. Because of this reduction of luminous intensity, the exposure must be increased. The filter factor, for a particular filter, light source, and photographic emulsion , is a measure of the required increase in exposure and is also a measure of the extent to which it reduces the quantity of light reaching the photographic material.

If the filter factor is F, the intensity of the light passing through it is inversely proportional to the filter factor or to 1/F. We may consider the effect of the filter, as well as that of the lens system, in determining the intensity of the image for the filter and lens system

        I' = IkT / F f² (M + 1)²                  (9)   


1) Careful investigations show that the reciprocity law is not exactly obeyed by photographic materials. Failure of the reciprocity law is not of serious consequence in most branches of practical photography, and, for a first approximation, may be neglected.

2) If the film is exposed intermittently, it is found that the photographic effects are not the same as when the film is exposed for the same time duration but continuously rather than intermittently.

from PP 211-214 of the HANDBOOK OF PHOTOGRAPHY by Henney and Dudley, McGraw Hill, 1939

Now aren't you glad you took the time to read this far??? How would you apply all this knowledge you have acquired to digital cameras and their sensors?? There must be a connection somewhere, don't you think??? andy

This article has been retrieved since January 15, 2002